How do I create a vertex at the intersecting point of two intersecting edges without addon or scripts, no faces involved?
If doing that without addons you need to subdivide both edges and use snapping tools to snap vertex from one of the edges to another. 

In ideal situation, your edges will be aligned with the world axes. However this is not always the case, hence this solution assumes they aren't (it still works even if they are) and uses custom transform orientation for snapping vertices. Alternatively it's possible to align view to the selected edge and use View transform orientation, although it might be less convenient navigating in the viewport. 


With edge selected, create Custom Transform Orientation with Ctrl+Alt+Space (or from the Properties shelf > Transform Orientations rollout). Note that Selection mode should be set to Edge so transform orientation is created for it. By default this transform orientation will be selected as current one; if not, choose it in the Transform Orientations menu or with Alt+Space.


Enable Snap during transform tool. Set Type of Element to snap to to Vertex (or Edge depending on which element you will hover over to snap vertex to). Set Pivot point to Active Element to use active selection as pivot.


Subdivide one of the edges once (W > Subdivide with edge selected).
Select vertex created after subdividing, then Shift select corner vertex of the edge to scale to. Activate Scale tool with S to move the vertex, press adjacent axis name twice to use custom transform axes and hover mouse over the vertex of the edge to snap to (or edge itself depending on what you selected in step 2).


Do the same for the other edge.


Select all and use W > Remove Doubles to join 2 vertices into one vertex at the intersection.